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User talk:Ikosarakt1/Buchholz hydras
According to the Buchholz hydra article, every child of root must have label 0 (or () in brackets). So you can't use [] or {} at the outer-most area. You can, however, use ([]) or ({}). -- I want more 12:57, April 18, 2013 (UTC) I agree. Hydra [] is irreducible, as is almost every hydra on list. I don't think adding () pair around every root child would make ordinals correct, but this is best idea I have. LittlePeng9 (talk) 13:13, April 18, 2013 (UTC) It should be confusing, in that case ([])([])([]) can correspond either to \(\varepsilon_0*2\) or \(\eta_0\). Why [] is irreducible? I believe that we can expand it as (((...((()))...))) (with number of steps+1 ()'s), which is quite logical. Ikosarakt1 (talk ^ ) 13:17, April 18, 2013 (UTC) Our argument is Buchholz's definition. Hydra [] would be vertex with label 1 connected to root with label +. In order to reduce that vertex we look below it to find vertex with label <1. But there is no such vertex, so we can't reduce this vertex. That's why every root child has label 0 in original definition. Of course, you can use other cut-rules, but you need your own definitions. LittlePeng9 (talk) 14:24, April 18, 2013 (UTC) I believe that we can use (([]))(([]))([]) for \(\varepsilon_0*2\) and ([])([])([]) for \(\eta_0\) in order to avoid this ambiguousness. Ikosarakt1 (talk ^ ) 14:35, April 18, 2013 (UTC) Also, there is another problem in computation - by this paper by Buchholz deleting >0 label doesn't depend on chosen n. That means hydra ([]) will always reduce to (((()))), which is \(\omega^\omega\). But there are formulations of labelled hydra games in which we stack resulting trees on top of each other n times. I think this is what better describes your list. LittlePeng9 (talk) 15:34, April 18, 2013 (UTC) I think Buchholz made a mistake labelling the root node + and all the children nodes 0. He should have just labelled the root node 0, that works just fine. If we assume the root node lies underneath every string, e.g. [] is a tree with a root node connected to a node labelled 1, then it would be legal (under the new definition). LittlePeng9 is right that the the ordinals are not correct under the standard rules - I think that you should say something about replacing the standard rule for u+1, which just stacks one tree, with a new rule that stacks n trees. Deedlit11 (talk) 17:37, April 18, 2013 (UTC) If [] is epsilon_0, how can []() be epsilon_1? () is a leaf that dies after one chop. Also, I don't see how we can have [][] > []([]) > ([])[]. Deedlit11 (talk) 00:52, April 19, 2013 (UTC) How these hydras should be reduced? I just believed that I can simulate theta function and Bird's separators using that. For example, [] corresponds to his backslash separator, when we have leaves to the left and right, it increments corresponding separator to level that assignes by current hydra. So, if () adds 1 and [] => \ 2, then []() => \ 3, and expands to ((...([])[])[])...)[] (with number of steps+1 []'s). By that, [][] indicates two backslashes together and has level \ 1 \ 2, while []([]) => \ 1 [1 \ 2 2] and ([])[] => {n,n \ 2 3}. Ikosarakt1 (talk ^ ) 09:13, April 19, 2013 (UTC) () pair reduces exactly like in Kirby-Paris hydra - we create n copies of subtree and we append it to same vertex. Cutting () doesn't increase hydra's height. LittlePeng9 (talk) 10:40, April 19, 2013 (UTC) Never continue? Jiawhein \(a\)\(l\) 10:13, May 9, 2013 (UTC) If [] is epsilon(0) and {} is BHO, why <> is theta(epsilon_{Omega_2+1}) and || is theta(epsilon_{Omega_3+1})? AarexTiao 22:15, September 3, 2013 (UTC) :This is because Buchholz's hydras diagonalize through system ID_w of repeated inductive definitions, and mentioned ordinals are strengths of ID_n for n=0,1,2,3, respectively. LittlePeng9 (talk) 08:39, September 4, 2013 (UTC) Error ([]) expands to ((())) and should have level w. (()) has level e0. Wythagoras (talk) 06:23, October 17, 2014 (UTC) ([]) is the first legal hydra which can't be expressed only with ()'s. Using ()'s, we can express all ordinals below e(0). So ([]) has order type e(0). Ikosarakt1 (talk ^ ) 07:57, October 17, 2014 (UTC)